Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities
aa r X i v : . [ m a t h . A P ] F e b Singular doubly nonlocal elliptic problems with Choquard typecritical growth nonlinearities
Jacques Giacomoni ∗ , Divya Goel † , and K. Sreenadh ‡ Universit´e de Pau et des Pays de l’Adour, LMAP (UMR E2S-UPPA CNRS 5142)Bat. IPRA, Avenue de l’Universit´e F-64013 Pau, France Department of Mathematics, Indian Institute of Technology Delhi,Hauz Khaz, New Delhi-110016, India
Abstract
The theory of elliptic equations involving singular nonlinearities is well studied topicbut the interaction of singular type nonlinearity with nonlocal nonlinearity in ellipticproblems has not been investigated so far. In this article, we study the very singularand doubly nonlocal singular problem ( P λ )(See below). Firstly, we establish a very weakcomparison principle and the optimal Sobolev regularity. Next using the critical pointtheory of non-smooth analysis and the geometry of the energy functional, we establishthe global multiplicity of positive weak solutions. Key words : Choquard equation, Fractional Laplacian, Singular Nonlinearity, Non smoothanalysis, Regularity. ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] The purpose of the article is to discuss the existence and multiplicity of weak solutions to thefollowing singular problem:( P λ ) ( − ∆ ) s u = u − q + λ Z Ω | u | ∗ µ ( y ) | x − y | µ dy ! | u | ∗ µ − u, u > Ω,u = 0 in R N \ Ω, for all q > , N ≥ s, s ∈ (0 , , ∗ µ = N − µN − s and Ω is a bounded domain in R N with smoothboundary. Here the operator ( − ∆ ) s is the fractional Laplacian defined as( − ∆ ) s u ( x ) = − P.V. Z R N u ( x ) − u ( y ) | x − y | N +2 s dy where P.V denotes the Cauchy principal value.The problems involving singular nonlinearity have a very long history. In the pioneeringwork [12], Crandall, Rabinowitz and Tartar [12] proved the existence of a solution of classicalelliptic PDE with singular nonlinearity using the approximation arguments. Later manyresearchers studied the problems involving singular nonlinearity. Haitao [26] studied thefollowing problem − ∆u = au − q + bu N +2 N − , u > Ω, u = 0 on ∂Ω (1.1)where Ω ⊂ R N ( N ≥
3) is a smooth bounded domain. If a = λ and b = 1, and q ∈ (0 , q ∈ (0 , a = λ and b = 1, and q ∈ (0 , λ .While in [29], authors studied the problem for all q > a = 1 b = λ , and established a globalmultiplicity result using the nonsmooth analysis. For more details on singular type problems,we refer to [11, 18, 23, 24, 26, 27] and references therein.The study of nonlinear elliptic problems with critical terms motivated by Hardy-Littlewood-Sobolev inequality started long back and attracted lot of researchers due to its wide applica-tions. Indeed, it was originated in the framework of various physical models. One of the firstapplications was found in H. Fr¨ohlich and S. Pekar model of the polaron, where free electronsin an ionic lattice interact with photons associated to the deformations of the lattice or withthe polarization that it creates on the medium [15, 16]. In the modeling of one componentplasma, Ph. Choquard gave the model which involves Choquard equation [30]. Later on suchnonlinear problems are called Choquard equations and many researchers studied these typeof problems to understand the existence, uniqueness, radial symmetry and regularity of thesolutions [33, 34, 35] and references therein. Pertaining the Choquard type critical exponentproblems on bounded domains, Gao and Yang [17] studied the Brezis-Nirenberg type exis-tence and nonexistence results with Choquard critical nonlinearities. In [9], [37] and[41] arestudied a Brezis-Nirenberg type problem with uppercritical growth, concentration profiles ofground states and existence of semiclassical states respectively.Nonlocal problems involving fractional Laplacian challenged a lot of researchers due to thelarge spectrum of applications. Consider the following problem( − ∆ ) s u = f ( x, u ) in Ω, u = 0 in R N \ Ω (1.2)where f is a Carath´eodory function. The questions of existence, multiplicity and regularityof solutions to problem (1.2) have been extensively studied in [32, 1] and references therein.Concerning the existence and multiplicity of solutions to doubly nonlocal problems, a lot ofworks have been done. For a detailed state of art, one can refer [10, 13, 36] and referencestherein.On the other hand, Barrios et al. [4] started the work on nonlocal equations with singularnonlinearity. Precisely, [4] deal with the existence of solutions to the following problem( − ∆ ) s u = λ (cid:18) a ( x ) u r + f ( x, u ) (cid:19) in Ω, u = 0 in R N \ Ω (1.3)where Ω is a bounded domain with smooth boundary, N > s, < s < , r, λ > , f ( x, u ) ∼ u p , < p < ∗ s −
1. In the spirit of [12], here authors first prove the existence of solutions u n tothe equation with singular term 1 /u r replaced by 1 / ( u + 1 /n ) r and use the uniform estimateson the sequence { u n } to finally prove the existence of a solution to (1.3). Furthermore,authors prove some Sobolev regularity, in particular for r > u r +12 ∈ X . In case of s = 1, optimal Sobolev regularity was established in [5] and [6] for semilinear and quasilinearelliptic type problems respectively. But in case of 0 < s <
1, the question of optimal Sobolevregularity still remained an open question. The regularity issue is of independent interest. Inthe recent times, Adimurthi, Giacomoni and Santra [2] studied the problem (1.3) with f = 0and complement the results of [4]. In particular they obtained the boundary behaviour andH¨older regularity of the classical solution. Then exploiting this asymptotic behavior, authorsobtained multiplicity of classical solutions by global bifurcation method in the framework ofweighted spaces for (1.3) with subcritical f .Regarding the critical case, Giacomoni, Mukherjee and Sreenadh [21] studied the problem(1.3) with a = 1 /λ, r > , and f ( x, u ) ∼ u ∗ s − . Here authors extended the techniques of [29]in fractional framework and proved the existence and multiplicity of solutions in C α loc ( Ω ) ∩ L ∞ ( Ω ) for some α >
0. Recently, authors [20] proved the global multiplicity result for (1.3)with a = 1 /λ, p = 2 ∗ s − r (2 s − < (1 + 2 s ) for energy solutions. Concerning thedoubly nonlocal problem with singular operators, in [19], we studied the regularity results forthe problems of th type ( P λ ) with 0 < q <
1. But the questions of existence, multiplicityof solutions to the problem ( P λ ) was a completely open problem even when s = 1. Alsothe question of (H¨older, Sobolev) regularity of solutions for q ≥ H sloc phenomena. In case of s = 1,this result was established in [8] with a slightly different approach. We first prove the L ∞ ( Ω )estimate for solutions of ( P λ ) by establishing the relation between the solutions of ( P λ ) and( f P λ ) (See Section 2). The techniques used here can be applied in a more general contextand are of independent interest. Next, using the results of [2] and Lemma 3.1 we prove theasymptotic behavior and optimal H¨older and Sobolev regularity of weak solutions.In this paper we have given a consolidated approach to prove the global multiplicity result forthe problem ( P λ ) exploiting convex properties of the singular nonlinearity and the geometryof the energy functional. To the best of our knowledge there is no previous contribution whichdeals the Choquard problem with singular nonlinearity. Further, the results proved in thisarticle are new and novel even in case of s = 1 where the approach can be closely adapted.For simplicity of illustration, we set some notations. We denote k u k L p ( Ω ) by | u | p , k u k X by k u k , [ u ] H s ( A ) = R A R A ( u ( x ) − u ( y )) | x − y | N +2 s dxdy . The positive constants C, c , c · · · values change caseby case.Turning to the paper organization: In Section 2, we define the function spaces, give somepreliminaries of nonsmooth analysis and further state the main results of the article. InSection 3, we establish a very weak comparison principle. In Section 4, we established theregularity of solutions to ( P λ ). In sections 5 and 6 , we prove the existence of first and secondsolution to ( P λ ). We recall the Hardy-Littlewood-Sobolev Inequality which is foundational in study of Choquardproblems of the type ( P λ ) Proposition 2.1. [31] Let t, r > and < µ < N with /t + µ/N + 1 /r = 2 , f ∈ L t ( R N ) and h ∈ L r ( R N ) . There exists a sharp constant C ( t, r, µ, N ) independent of f, h , such that Z R N Z R N f ( x ) h ( y ) | x − y | µ dydx ≤ C ( t, r, µ, N ) | f | t | h | r . Consider the space X := { u ∈ H s ( R N ) : u = 0 a.e in R N \ Ω } , equipped with the inner product h u, v i = Z Z Q ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | N +2 s dxdy. where Q = R N \ ( Ω c × Ω c ). From the embedding results ([32]), the space X is continuouslyembedded into L r ( R N ) with r ∈ [1 , ∗ s ] where 2 ∗ s = NN − s . The embedding is compact if andonly if r < ∗ s . The best constant S of the classical Sobolev embedding is defined S = inf u ∈ X \{ } R R N R R N | u ( x ) − u ( y ) | | x − y | N +2 s dxdy (cid:0)R Ω | u | ∗ s (cid:1) / ∗ s . Consequently, we define S H,L = inf u ∈ X \{ } R R N R R N | u ( x ) − u ( y ) | | x − y | N +2 s dxdy (cid:18)R R N R R N | u | ∗ µ ( x ) | u | ∗ µ ( y ) | x − y | µ dxdy (cid:19) / ∗ µ . Lemma 2.2. [36] The constant S H,L is achieved if and only if u = C (cid:18) bb + | x − a | (cid:19) N − s where C > is a fixed constant , a ∈ R N and b ∈ (0 , ∞ ) are parameters. Moreover, S = S H,L ( C ( N, µ )) N − s N − µ . Definition 2.3.
For φ ∈ C ( Ω ) with φ > in Ω , the set C φ ( Ω ) is defined as C φ ( Ω ) = { u ∈ C ( Ω ) : there exists c ≥ such that | u ( x ) | ≤ cφ ( x ) , for all x ∈ Ω } , endowed with the natural norm (cid:13)(cid:13)(cid:13)(cid:13) uφ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( Ω ) . Definition 2.4.
The positive cone of C φ ( Ω ) is the open convex subset of C φ ( Ω ) defined as C + φ ( Ω ) = (cid:26) u ∈ C φ ( Ω ) : inf x ∈ Ω u ( x ) φ ( x ) > (cid:27) . The barrier function φ q is defined as follows: φ q = φ if 0 < q < ,φ (cid:16) log (cid:16) Aφ (cid:17)(cid:17) if q = 1 ,φ q +1 if q > , where φ is the normalized ( k φ k L ∞ ( Ω ) = 1) eigenfunction corresponding to the smallesteigenvalue of ( − ∆ ) s on X and A > diam( Ω ). We recall that φ ∈ C s ( R N ) and φ ∈ C + d s ( Ω )(See Proposition 1.1 and Theorem 1.2 of [40]).Before giving the definition of weak solution to ( P λ ) we discuss the solution of the followingpurely singular problem( − ∆ ) s u = u − q , u > Ω, u = 0 in R N \ Ω. (2.1)From [2] we know the following result. Proposition 2.5.
Let q > . Then there exists u ∈ L ∞ ( Ω ) ∩ C + φ q classical solution to (2.1) .Moreover, u has the following properties:(i) u ∈ X if and only if q (2 s − < (2 s + 1) and in this case we have unique classicalsolution to (2.1) .(ii) u ∈ C γ ( R N ) where γ = s if q < ,s − ε if q = 1 , for all ε > small enough , q +1 if q > . (2.2) Remark 2.6.
We remark that since u ∈ L ∞ ( Ω ) ∩ C + φ q ∩ C γ ( R N ) . So we can achieve theinterior C ∞ regularity. That is for any compact set Ω ′ ⊂ Ω we have u ∈ C ∞ ( Ω ′ ) . From thisone can easily prove the fact that u ∈ H s loc ( Ω ) . Lemma 2.7. (a) If q (2 s − ≥ (2 s + 1) then u γ ∈ X if and only if γ > (2 s − q +1)4 s . Moreover the lower bound on γ is optimal in the sense that u γ X if γ ≤ (2 s − q +1)4 s .(b) ( u − ε ) + ∈ X for all ε > . Proof. (a) Let ξ ( x ) = x γ , γ >
1. Observe that ξ is convex and differentiable function on R + . Hence using this and the fact that φ ∈ C + d s ( Ω ) and u ∈ C + φ q , we deduce that k ξ ( u ) k = h ( − ∆ ) s ξ ( u ) , ξ ( u ) i ≤ Z Ω ξ ( u ) ξ ′ ( u )( − ∆ ) s u dx = Z Ω γu γ − q − dx ≤ C Z Ω d s (2 γ − q − q +1 dx. We know that d s (2 γ − q − q +1 ∈ L ( Ω ) if and only if γ > (2 s − q +1)4 s . This settles first part of theproof. For the second part, let γ ≤ (2 s − q +1)4 s and if possible let u γ ∈ X . Consider Z Ω u γ d s dx ≤ C Z Ω d sγq +1 − s dx = ∞ It contradicts the fact that u γ ∈ X and then satisfies the Hardy inequality.(b) Let A = { x : u ( x ) > ε } then k ( u − ε ) + k ≤ Z A Z A | u ( x ) − u ( y ) | | x − y | N +2 s dxdy + 2 Z R N \ A Z A | u ( x ) − u ( y ) | | x − y | N +2 s dxdy ≤ κ Z A Z A | u γ ( x ) − u γ ( y ) | | x − y | N +2 s dxdy + 2 κ Z R N \ A Z A | u γ ( x ) − u γ ( y ) | | x − y | N +2 s dxdy ≤ ( κ + κ ) Z Q | u γ ( x ) − u γ ( y ) | | x − y | N +2 s dxdy < ∞ . By the mean value theorem and convexity arguments, one can easily prove the existence of κ , κ such that κ , κ ≥ ε γ − . Hence the proof is complete. (cid:3) The energy functional associated to the probelm ( P λ ) is I ( u ) = 12 Z Q | u ( x ) − u ( y ) | | x − y | N +2 s dxdy − − q Z Ω | u | − q dx − λ ∗ µ Z Z Ω × Ω | u | ∗ µ | u | ∗ µ | x − y | µ dxdy. Though the functional I is continuous on X when 0 < q < q ≥
1, the functional I isnot even finite at all points of X . Also I it can be shown that I is not Gˆateaux differentiableat all points of X . The doubly nonlocal nature of the problem ( P λ ) and the lack of regularityof I force to use to introduce a quite general definition of weak solution. The Lemma 2.7motivates the following definition of weak solution to the problem ( P λ ). Definition 2.8.
A function u ∈ H s loc ( Ω ) ∩ L ∗ s ( Ω ) is said to be a weak solution of ( P λ ) if the following hold:(i) there exists m K > such that u > m K for any compact set K ⊂ Ω .(ii) For any φ ∈ C ∞ c ( Ω ) , h u, φ i = Z Ω u − q φ dx + λ Z Z Ω × Ω u ∗ µ ( x ) u ∗ µ − ( y ) φ ( y ) | x − y | µ dxdy. (iii) ( u − ε ) + ∈ X for all ε > . Lemma 2.9.
Let u be a weak solution to ( P λ ) as it is defined in Definition 2.8. Then for allcompactly supported ≤ v ∈ X ∩ L ∞ ( Ω ) , we have h u, v i − Z Ω u − q v dx − Z Z Ω × Ω u ∗ µ u ∗ µ − v | x − y | µ dxdy = 0 . (2.3) Proof.
Let 0 ≤ v ∈ X ∩ L ∞ ( Ω ) be compactly supported function. Then there exists asequence v n ∈ C ∞ c ( Ω ) such that v n ≥ , K := ∪ n supp v n is contained in compact set of Ω, {| v n | ∞ } is bounded sequence and v n → v strongly in X . Since u is a weak solution to( P λ ), we have h u, v n i − Z Ω u − q v n dx − Z Z Ω × Ω u ∗ µ u ∗ µ − v n | x − y | µ dxdy = 0 . (2.4)Consider Z Q ( u ( x ) − u ( y ))( v n ( x ) − v n ( y )) | x − y | N +2 s dxdy = Z K Z K ( u ( x ) − u ( y ))( v n ( x ) − v n ( y )) | x − y | N +2 s dxdy + 2 Z Ω \ K Z K ( u ( x ) − u ( y )) v n ( x ) | x − y | N +2 s dxdy + 2 Z R N \ Ω Z Ω u ( x ) v n ( x ) | x − y | N +2 s dxdy := I + II + III.
Now using the fact that u ∈ H s loc ( Ω ) and the strong convergence of v n , as n → ∞ , we obtain I → Z K Z K ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | N +2 s dxdy. Next taking into account u ∈ L ∗ s ( Ω ) and the fact that v n is uniformly bounded in L ∞ ( Ω ),by dominated convergence theorem, we deduce that II → Z Ω \ K Z K ( u ( x ) − u ( y )) v ( x ) | x − y | N +2 s dxdy. Similarly,
III → R R N \ Ω R Ω u ( x ) v ( x ) | x − y | N +2 s dxdy . Hence h u, v n i → h u, v i as n → ∞ . Trivially R Ω u − q v n dx → R Ω u − q v dx as n → ∞ . Also using the strong convergence of sequence v n andthe fact that u ∈ L ∗ s ( Ω ), we infer that Z Z Ω × Ω u ∗ µ u ∗ µ − v n | x − y | µ dxdy → Z Z Ω × Ω u ∗ µ u ∗ µ − v | x − y | µ dxdy. It implies that passing the limit as n → ∞ in (2.4), we have (2.3) for all compactly supported0 ≤ v ∈ X ∩ L ∞ ( Ω ). (cid:3) In the direction of existence of solution to ( P λ ), we translate the problem ( P λ ) by the solution u of problem (2.1). Consider the translated problem( f P λ ) ( − ∆ ) s u + u − q − ( u + u ) − q = λ Z Ω ( u + u ) ∗ µ ( y ) | x − y | µ dy ! ( u + u ) ∗ µ − , u > Ω,u = 0 in R N \ Ω. Observe that u + u is a solution to ( P λ ) if and only if u ∈ X is a solution to ( f P λ ). Definethe function g : Ω × R → R ∩ {∞} as g ( x, s ) = ( u − q − ( s + u ) − q if s + u > , −∞ otherwiseand G ( x, y ) = R y g ( x, τ ) dτ for ( x, y ) ∈ Ω × R . Definition 2.10.
A function u ∈ X is a subsolution (resp. supersolution) to ( f P λ ) if thefollowing holds(i) u + ∈ X (resp. u − ∈ X );(ii) g ( · , u ) ∈ L loc ( Ω ) ;(iii) For all φ ∈ C ∞ c ( Ω ) , φ ≥ , we have h u, φ i + Z Ω g ( x, u ) φ dx − Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − φ | x − y | µ dxdy ≤ resp. ≥ . Definition 2.11.
A function u ∈ X is a weak solution to ( f P λ ) if it is both sub and superso-lution and u ≥ in Ω . Lemma 2.12.
Let u ∈ X be a weak solution to ( f P λ ) . Then for any v ∈ X , we have h u, v i + Z Ω g ( x, u ) v dx − Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − v | x − y | µ dxdy = 0 . (2.5) Proof.
Let 0 ≤ v ∈ X then by [21, Lemma 3.1], there exists an increasing sequence0 { v n } ∈ X such that v n has a compact support and v n → v strongly in X . For each n , thereexists a sequence φ kn ∈ C ∞ c ( Ω ) such that φ kn ≥ , ∪ k supp φ kn is contained in compact set of Ω, {| φ kn | ∞ } is bounded sequence and φ kn → v n strongly in X . Since u is a weak solution to( f P λ ), we have h u, φ kn i + Z Ω g ( x, u ) φ kn dx − Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − φ kn | x − y | µ dxdy = 0 . Using the fact that φ kn → v n strongly in X as n → ∞ , we deduce that h u, v n i + Z Ω g ( x, u ) v n dx − Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − v n | x − y | µ dxdy = 0 . Now by using the dominated convergence theorem and the strong convergence of the sequence v n in X , we get g ( x, u ) v ∈ L ( Ω ) and we have (2.5) for any 0 ≤ v ∈ X . For any v ∈ X , v = v + − v − . Employing the above procedure for v + and v − separately, we obtain the desiredresult. Hence the proof. (cid:3) With this functional framework we record now the statement of our main Theorems.
Theorem 2.13.
Let µ ≤ min { s, N } . There exists a Λ > such that1. For every λ ∈ (0 , Λ ) the problem ( P λ ) admits two solutions in C + φ q ( Ω ) ∩ L ∞ ( Ω ) .2. For λ = Λ there exists a solution in C + φ q ( Ω ) ∩ L ∞ ( Ω ) .3. For λ > Λ , there exists no solution.Moreover, solution belongs to X if and only if q (2 s − < (2 s + 1) . Concerning the H¨older and Sobolev regularity of solutions we have the following Theorem.
Theorem 2.14.
Let µ ≤ min { s, N } . Let q > , λ ∈ (0 , Λ ) . Then any weak solutionin the sense of Definition 2.8 is classical and belongs to C γ ( R N ) where γ is defined (2.2) .Furthermore any weak solution satisfies the statements of Lemma 2.7. Remark 2.15.
We point out that regularity results contained in Theorem 2.14 are muchstronger as compared to those obtained in [2, 20] where regularity of continuous solutions areonly investigated.
In this subsection we record some basic definitions, observations and linking theorem tononsmooth functionals. We remark that in case of q (2 s − < (2 s + 1), one can adapt thevariational techniques of the [26, 20] to prove the global multiplicity result as in Theorem 2.13but to incorporate the case of q large we adopt the following notions of non-smooth analysis.1 Definition 2.16.
Let H be a Hilbert space and J : H → ( −∞ , ∞ ] be a proper (i.e. J
6≡ ∞ )lower semicontinuous functional.(i) Let D ( J ) = { u ∈ H : J ( u ) < ∞} be the domain of J . For every u ∈ D ( J ) , we definethe Fr´echet sub-differential of J at u as the set ∂ − J ( u ) = (cid:26) z ∈ H : lim v → u J ( v ) − J ( u ) − h z, v − u ik v − u k H ≥ (cid:27) . (ii) For each u ∈ H , we define (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ − J ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( min {k z k H : z ∈ ∂ − J ( u ) } if ∂ − J ( u ) = ∅ , ∞ if if ∂ − J ( u ) = ∅ . We know that ∂ − J ( u ) is a closed convex set which may be empty. If u ∈ D ( J ) is a localminimizer for J , then it can be seen that ∈ ∂ − J ( u ) . Remark 2.17.
We remark that if J : H → ( −∞ , ∞ ] be a proper, lower semicontinuous,convex functional, J : H → R is a C functional and J = J + J , then ∂ − J ( u ) = ∇ J ( u ) + ∂J ( u ) for every u ∈ D ( J ) = D ( J ) , where ∂J denotes the usual subdifferential of the convexfunctional J . Thus, u is said to be a critical point of J if u ∈ D ( J ) and for every v ∈ H ,we have h∇ J ( u ) , v − u i + J ( v ) − J ( u ) ≥ . Definition 2.18.
For a proper, lower semicontinuous functional J : H → ( −∞ , ∞ ] , we saythat J satisfies Cerami’s variant of the Palais-Smale condition at level c (in short, J satisfies ( CPS ) c , if any sequence { z n } ⊂ D ( J ) such that J ( z n ) → c and (1 + z n ) ||| ∂ − J ( z n ) ||| → hasa strongly convergent subsequence in H . Analogous to the mountain pass theorem, we have the following linking theorem for non-smooth functionals.
Theorem 2.19. [39] Let H be a Hilbert space. Assume J = J + J , where J : H → ( −∞ , ∞ ] is a proper, lower semicontinuous, convex functional and J : H → R is a C -functional. Let B N , S N − denote the closed unit ball and its boundary in R N respectively. Let ϕ : S N − → D ( J ) be a continuous function such that Σ = { ψ ∈ C ( B N , D ( J )) : ψ | S N − = ϕ } 6 = ∅ . Let A be a relatively closed subset of D ( J ) such that A ∩ ϕ ( S N − ) = ∅ , A ∩ ϕ ( B N ) = ∅ for all ψ ∈ Σ and inf J ( A ) ≥ sup J ( ϕ ( S N − )) . Define c = inf ψ ∈ Σ sup x ∈ B N J ( ψ ( x )) . Assume that c is finite and that J satisfies ( CPS ) c ) .Then there exists u ∈ D ( J ) such that J ( u ) = c and ∈ ∂ − J ( u ) . Furthermore, if inf J ( A ) = c ,then there exists u ∈ A ∩ D ( J ) such that J ( u ) = c and ∈ ∂ − J ( u ) . Here we establish a new weak comparison principle that can be applied in the setting of H sloc ( Ω ) sub and supersolutions to ( P λ ) and cover all q > q (2 s − < s + 1 is required). Lemma 3.1.
Let F ∈ X ∗ and let z, w ∈ H s loc ( Ω ) be such that z, w > a.e in Ω, z, w ≥ ∈ R N , z − q , w − q ∈ L loc ( Ω ) , ( z − ε ) + ∈ X for all ε > , z ∈ L ( Ω ) and h z, φ i ≤ Z Ω z − q φ dx + ( F, φ ) , h w, φ i ≥ Z Ω w − q φ dx + ( F, φ ) (3.1) for all compactly supported φ ∈ X ∩ L ∞ ( Ω ) with φ ≥ . Then z ≤ w a.e in Ω . Proof.
Let us denote that Ψ n : R → R the primitive of the function s ( max {− s − q , − n } if s > , − n if s ≤ Ψ n (1) = 0. Let us define a proper lower semicontinuous, strictly convex functional˘ H ,n : L ( Ω ) → R given by˘ H ,n ( u ) = ( k u k + R Ω Ψ n ( u ) dx if u ∈ X , ∞ if u ∈ L ( Ω ) \ X . We define H ,n : L ( Ω ) → R as H ,n ( u ) = ˘ H ,n ( u ) − min ˘ H ,n = ˘ H ,n ( u ) − ˘ H ,n ( u ,n )where u ,n ∈ X is the minimum of ˘ H ,n . More generally, for F ∈ X ∗ we set:˘ H F,n ( u ) = ( ˘ H ,n ( u ) − ( F, u − u ,n ) if u ∈ X , ∞ if u ∈ L ( Ω ) \ X . Let ε > n > ε − q and let v be the minimum of the functional ˘ H F,n on the convex set K = { ϕ ∈ X : 0 ≤ ϕ ≤ w a.e in Ω } . Then for all ϕ ∈ K we get h v, ϕ − v i ≥ − Z Ω Ψ ′ n ( v )( ϕ − v ) dx + ( F, ϕ − v ) . (3.2)Let 0 ≤ ϕ ∈ C ∞ c ( Ω ) , t >
0. Define ϕ t := min { v + tϕ, w } . Now using the fact that w ∈ H s loc ( Ω ) , v ∈ X , ϕ ∈ C ∞ c ( Ω ), we have ϕ t ∈ X . Furthermore, ϕ t is uniformly boundedin X for all t <
1. For the proof let A = supp( ϕ ). Since on Ω \ A, ϕ t = v and otherwise v ≤ ϕ t ≤ w , we deduce that3 Z Q ( ϕ t ( x ) − ϕ t ( y )) | x − y | N +2 s dxdy ≤ [ ϕ t ] H s ( A ) + Z Ω \ A Z Ω \ A ( v ( x ) − v ( y )) | x − y | N +2 s dxdy + 2 Z Ω \ A Z A ( v ( x ) − v ( y )) + 2 tv ( y ) ϕ ( y ) + t ϕ ( y ) | x − y | N +2 s dxdy + 2 Z R N \ Ω Z Ω ( v + tϕ ( y )) | x − y | N +2 s dxdy < ∞ . (3.3)Employing the fact that for any g : R N → R , | g + ( x ) − g + ( y ) | ≤ | g ( x ) − g ( y ) | for all x, y ∈ R N coupled with ϕ t = v + tϕ − ( v + tϕ − w ) + , for all t <
1, we conclude that[ ϕ t ] H s ( A ) ≤ Z A Z A (( v + tϕ )( x ) − ( v + tϕ )( y )) | x − y | N +2 s dxdy + 2 Z A Z A (( v + tϕ − w )( x ) − ( v + tϕ − w )) | x − y | N +2 s dxdy ≤ C (cid:0) [ v ] H s ( A ) + [ ϕ ] H s ( A ) + [ w ] H s ( A ) (cid:1) . (3.4)From (3.3) and (3.4) we obtain that ϕ t is uniformly bounded in X . Take the subsequence(still denoted by ϕ t ) such that ϕ t ⇀ v weakly in X as t → + . Now test (3.2) with ϕ t , weget h v, ϕ t − v i ≥ − Z Ω Ψ ′ n ( v )( ϕ t − v ) dx + ( F, ϕ t − v ) . (3.5)Using (3.1) and the fact that w − q ≥ − Ψ ′ n ( w ), we infer that w satisfies h w, φ i ≥ − Z Ω Ψ ′ n ( w ) φ dx + ( F, φ ) (3.6)Deploying the fact that ϕ t ≤ w coupled with ϕ t − v − tϕ ≤ ϕ t = w then ϕ t − v − tϕ = 0,we deduce that Z Q ( ϕ t ( x ) − ϕ t ( y ))(( ϕ t − v − tϕ )( x ) − ( ϕ t − v − tϕ )( y )) | x − y | N +2 s dxdy ≤ Z Q w ( x )( ϕ t − v − tϕ )( x ) | x − y | N +2 s dxdy + Z Q w ( y )( ϕ t − v − tϕ )( y ) | x − y | N +2 s dxdy − Z Q w ( x )( ϕ t − v − tϕ )( y ) | x − y | N +2 s dxdy − Z Q w ( y )( ϕ t − v − tϕ )( x ) | x − y | N +2 s dxdy = h w, ϕ t − v − tϕ i . (3.7)Similarly, R Ω ( Ψ ′ n ( ϕ t ) − Ψ ′ n ( w ))( ϕ t − v − tϕ ) dx ≤ Ψ ′ n ( w ) ≤ − w − q . Taking into4account (3.1), (3.5), (3.6), (3.7) and above observations, we deduce that k ϕ t − v k − Z Ω ( − Ψ ′ n ( ϕ t ) + Ψ ′ n ( v ))( ϕ t − v ) dx = h ϕ t , ϕ t − v i + Z Ω Ψ ′ n ( ϕ t )( ϕ t − v ) dx − h v, ϕ t − v i − Z Ω Ψ ′ n ( v )( ϕ t − v ) dx ≤ h ϕ t , ϕ t − v i + Z Ω Ψ ′ n ( ϕ t )( ϕ t − v ) dx − ( F, ϕ t − v )= h ϕ t , ϕ t − v − tϕ i + Z Ω Ψ ′ n ( ϕ t )( ϕ t − v − tϕ ) dx − ( F, ϕ t − v − tϕ )+ t (cid:18) h ϕ t , ϕ i + Z Ω Ψ ′ n ( ϕ t ) ϕ dx − ( F, ϕ ) (cid:19) ≤ h w, ϕ t − v − tϕ i + Z Ω Ψ ′ n ( w )( ϕ t − v − tϕ ) dx − ( F, ϕ t − v − tϕ )+ t (cid:18) h ϕ t , ϕ i + Z Ω Ψ ′ n ( ϕ t ) ϕ dx − ( F, ϕ ) (cid:19) ≤ t (cid:18) h ϕ t , ϕ i + Z Ω Ψ ′ n ( ϕ t ) ϕ dx − ( F, ϕ ) (cid:19) . Therefore, we obtain that h ϕ t , ϕ i + Z Ω Ψ ′ n ( ϕ t ) ϕ dx − ( F, ϕ ) ≥ t (cid:18) k ϕ t − v k − Z Ω | Ψ ′ n ( ϕ t ) − Ψ ′ n ( v ) | ( ϕ t − v ) dx (cid:19) ≥ − Z Ω | Ψ ′ n ( ϕ t ) − Ψ ′ n ( v ) | ϕ dx. Now using the weak convergence of ϕ t and monotone convergence theorem, and dominatedconvergence theorem, we obtain h v, ϕ i ≥ − Z Ω Ψ ′ n ( v ) ϕ dx + ( F, ϕ ) . (3.8)Using the density argument, one can easily show that (3.8) is true for all ϕ ∈ X with ϕ ≥ Ω . Note that v ≥ z − ǫ − v ) + ⊂ supp( z − ǫ ) + ) that is, ( z − v − ε ) + ∈ X .So from (3.8), it implies that h v, ( z − v − ε ) + i ≥ − Z Ω Ψ ′ n ( v )( z − v − ε ) + dx + ( F, ( z − v − ε ) + ) . (3.9)Let ( z − v − ε ) + := g ∈ X such that 0 ≤ g ≤ z a.e in Ω . Let { ˆ g m } be a monotonicallyincreasing sequence in C ∞ c ( Ω ) such that { ˆ g m } converging to g in X and set g m = min { ˆ g + m , g } .Testing (3.1) with g m , we get h z, g m i ≤ Z Ω z − q g m dx + ( F, g m ) (3.10)5Observe that if g > z > ε . Now consider Z { g > } Z { g > } ( z ( x ) − z ( y ))(( g m − g )( x ) − ( g m − g )( y )) | x − y | N +2 s dxdy = Z { g > } Z { g > } (( z − ε ) + ( x ) − ( z − ε ) + ( y ))(( g m − g )( x ) − ( g m − g )( y )) | x − y | N +2 s dxdy ≤ k ( z − ε ) + k k ( g m − g ) k → m → ∞ . (3.11) Z R N \{ g > } Z { g > } ( z ( x ) − z ( y ))( g m ( x ) − g m ( y )) | x − y | N +2 s dxdy = Z R N \{ g > } Z { g > } (( z ( x ))( g m ( x )) | x − y | N +2 s dxdy − Z R N \{ g > } Z { g > } ( z ( y ))( g m ( x )) | x − y | N +2 s dxdy ≥ Z R N \{ g > } Z { g > } (( z ( x ))( g m ( x )) | x − y | N +2 s dxdy − Z R N \{ g > } Z { g > } ( z ( y ))( g ( x )) | x − y | N +2 s dxdy. (3.12)Taking into account the fact that z − q g m ≤ z − q g , (3.10), (3.11), (3.12) and monotone conver-gence theorem, if z − q g ∈ L ( Ω ) or z − q g L ( Ω ), we conclude that h z, g i ≤ Z Ω z − q g dx + ( F, g ) . That is, h z, ( z − v − ε ) + i ≤ Z Ω z − q ( z − v − ε ) + dx + ( F, ( z − v − ε ) + ) (3.13)Exploiting n ≥ ε − q , (3.9), (3.13), and the fact that for any measurable function h , h h + , h + i ≤h h , h + i , we obtain that h ( z − v − ε ) + , ( z − v − ε ) + i ≤ h z − v, ( z − v − ε ) + i≤ Z Ω ( z − q + Ψ ′ n ( v ))( z − v − ε ) + dx = Z Ω ( − Ψ ′ n ( z ) + Ψ ′ n ( v ))( z − v − ε ) + dx ≤ . Thus, z ≤ v + ε ≤ w + ε . Since ε was arbitrary chosen, hence proof follows. (cid:3) In this section, we start by extending some regularity results contained in [19] and concludethe proof of Theorem 2.14.
Lemma 4.1.
Any nonnegative solution to ( f P λ ) belongs to L ∞ ( Ω ) . Proof.
Let u ∈ X be any non negative weak solution to (4.6). Let u τ = min { u, τ } for τ >
0. Let φ = u ( u τ ) r − ∈ X ( r ≥
2) be a test function to problem ( f P λ ). Now from [19,Lemma 3.5], we have the following inequality4( r − r (cid:16) a | a k | r − − b | b k | r − (cid:17) ≤ ( a − b )( a k | a k | r − − b k | b k | r − ) . (4.1)where a, b ∈ R and r ≥
2. Using (4.1), we deduce that | u ( u τ ) r − | ∗ s ≤ C k u ( u τ ) r − k ≤ Cr r − Z Q ( u ( x ) − u ( y ))( φ ( x ) − φ ( y )) | x − y | N +2 s dxdy = Cr − g ( x, u ) u ( u τ ) r − dx + Z Ω Z Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − u ( u τ ) r − | x − y | µ dxdy ! ≤ Cr Z Ω Z Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − u ( u τ ) r − | x − y | µ dxdy ≤ Cr Z Ω Z Ω u ∗ µ u ∗ µ ( u τ ) r − | x − y | µ dxdy + Z Ω Z Ω u ∗ µ u ∗ µ − u ( u τ ) r − | x − y | µ dxdy + Z Ω Z Ω u ∗ µ u ∗ µ ( u τ ) r − | x − y | µ dxdy + Z Ω Z Ω u ∗ µ u ∗ µ − u ( u τ ) r − | x − y | µ dxdy ! ≤ Cr | u | ∗ µ ∗ s (cid:18)Z Ω ( u ∗ µ u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s | u | ∗ µ − ∞ (cid:18)Z Ω ( uu r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:18)Z Ω ( u ∗ µ u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s | u | ∗ µ − ∞ (cid:18)Z Ω ( uu r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s (4.2) Claim:
Let r = 2 ∗ s + 1. Then u ∈ L ∗ sr ( Ω ).In view of H¨older’s inequality, we have (cid:18)Z Ω ( u ∗ µ u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s = (cid:18)Z u ≤ R ( u ∗ µ u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + (cid:18)Z u>R ( | u | ∗ µ u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s ≤ R ∗ µ (cid:18)Z u ≤ R ( u r − τ ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + (cid:18)Z u>R ( uu r − τ ) ∗ s (cid:19) ∗ s (cid:18)Z u>R u ∗ s (cid:19) ∗ µ − ∗ s . (4.3)7Choose R > Z | u | >R | u | ∗ s dx ! ∗ s − ∗ s < Cr min | u | ∗ µ ∗ s , | u | ∗ µ ∗ s . (4.4)Taking into account (4.2), (4.3) jointly with (4.4), we obtain | u ( u τ ) r − | ∗ s ≤ Cr R ∗ µ | u | ∗ µ ∗ s (cid:18)Z u ≤ R ( u ∗ s − ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s | u | ∗ µ − ∞ (cid:18)Z Ω u ∗ s (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s R ∗ µ (cid:18)Z u ≤ R ( u ∗ s − ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ ∗ s | u | ∗ µ − ∞ (cid:18)Z Ω u ∗ s (cid:19) ∗ µ ∗ s . Appealing Fatou’s Lemma as τ → ∞ , we obtain || u | r | ∗ s ≤ Cr (cid:16) | u | ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:17) R ∗ µ (cid:18)Z u ≤ R ( u ∗ s − ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ − ∞ (cid:18)Z Ω u ∗ s (cid:19) ∗ µ ∗ s < ∞ . This establishes the Claim. Now let τ → ∞ in (4.2) and using the inequality x p < x for p < x ≥ || u | r | ∗ s ≤ Cr (cid:16) | u | ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:17) (cid:18)Z Ω ( u ∗ µ + r − ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s + | u | ∗ µ − ∞ (cid:18)Z Ω ( u r − ) ∗ s ∗ µ (cid:19) ∗ µ ∗ s ≤ Cr (cid:16) | u | ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:17) (cid:18)(cid:18) Z Ω ( u ∗ µ + r − ) ∗ s ∗ µ (cid:19) + | u | ∗ µ − ∞ (cid:18) Z Ω ( u r − ) ∗ s ∗ µ (cid:19)(cid:19) ≤ Cr (1 + | u | ∗ µ − ∞ + | Ω | ) (cid:16) | u | ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:17) (cid:18) Z Ω ( | u | r +2 ∗ µ − ) ∗ s ∗ µ dx (cid:19) . It implies (cid:18) Z Ω | u | ∗ sr dx (cid:19) ∗ ( r − ≤ C r − r (cid:18) Z Ω ( u ∗ µ − r ) ∗ s ∗ µ dx (cid:19) r − (4.5)where C r = 4 Cr (1 + | u | ∗ µ ∗ s + | Ω | ) (cid:16) | u | ∗ µ ∗ s + | u | ∗ µ ∗ s (cid:17) . For j ≥ r j +1 inductively as( r j +1 + 2 ∗ µ −
2) 2 ∗ s ∗ µ = 2 ∗ s r j . That is, ( r j +1 −
2) = (cid:16) ∗ µ (cid:17) j ( r − C r j +1 = 4 Cr j +1 (1 + | u | ∗ µ ∗ s + | Ω | ), it8follows that (cid:18) Z Ω | u | ∗ srj +12 dx (cid:19) ∗ s ( rj +1 − ≤ C rj +1 − r j +1 (cid:18) Z Ω ( u ∗ µ − r j ) ∗ s ∗ µ dx (cid:19) ∗ s ( rj − . Defining A j := (cid:18) R Ω ( u ∗ µ − r j ) ∗ s ∗ µ dx (cid:19) ∗ s ( rj − . Then by Claim and limiting argument,there exists C > A j +1 ≤ j +1 Y k =2 C (1 / r k − k A ≤ C A . Hence | u | ∞ ≤ C A . That is u ∈ L ∞ ( Ω ). (cid:3) Remark 4.2.
We remark that if u ∈ X be any weak solution of the following problem ( − ∆ ) s u = f ( x, u ) + Z Ω | u | ∗ µ ( y ) | x − y | µ dy ! | u | ∗ µ − in Ω, u = 0 in R N \ Ω, (4.6) where | f ( x, u )) | ≤ C (1 + | u | ∗ − ) and µ ≤ min { s, N } . Then by using the same assertions asin Lemma 4.1, we obtain that u ∈ L ∞ ( Ω ) . This complements in the singular case previousresults proved in [19]. Lemma 4.3.
Let z ∈ L ∗ s ( Ω ) be a positive function, let h ( x, z ) = Z Ω z ∗ µ ( y ) | x − y | µ dy ! z ∗ µ − .Assume u ∈ X be a positive weak solution to ( − ∆ ) s u + g ( x, u ) = h ( x, z ) in Ω, u = 0 in R N \ Ω. (4.7) Then ( u + u − ε ) + ∈ X for every ε > . Proof.
Using the assertions and arguments used in [21, Lemma 3.4], one can easily proofthe result, we leave it for the readers. (cid:3)
Lemma 4.4.
Let λ > and let z ∈ H s loc ( Ω ) ∩ L ∗ s ( Ω ) be a weak solution to ( P λ ) as it isdefined in definition 2.8. Then z − u is a positive weak solution to ( f P λ ) belonging to L ∞ ( Ω ) . Proof.
Consider problem (4.7) with z given. Then 0 is a strict subsolution to (4.7). Definethe functional I : X → ( −∞ , ∞ ] by I ( u ) = k u k + R Ω G ( x, u ) dx − λ ∗ µ RR Ω × Ω z ∗ µ z ∗ µ − u | x − y | µ dxdy if G ( · , u ) ∈ L ( Ω ) , ∞ otherwise.Moreover for the closed convex set K = { u ∈ X : u ≥ } we define I K : X → ( −∞ , ∞ ]9by I K ( u ) = ( I ( u ) if u ∈ K and G ( · , u ) ∈ L ( Ω ) , ∞ otherwise.we can easily prove that there exists u ∈ K such that I K ( u ) = inf I K ( K ). It implies that0 ∈ ∂ − I K ( u ). Now from Proposition 5.2, we obtain that u is a non negative solution to(4.7). Using the Lemma 4.3, Lemma 2.9 and assertions as in Lemma 2.12, we obtain that( u + u − ε ) + ∈ X for every ε > h u + u, v i − Z Ω ( u + u ) − q v dx − Z Z Ω × Ω z ∗ µ z ∗ µ − v | x − y | µ dxdy = 0 h z, v i − Z Ω z − q v dx − Z Z Ω × Ω z ∗ µ z ∗ µ − v | x − y | µ dxdy = 0for all compactly supported 0 ≤ v ∈ X ∩ L ∞ ( Ω ). To prove the above equations for allcompactly supported 0 ≤ v ∈ X ∩ L ∞ ( Ω ) one can use the fact that u ∈ X , u ∈ H s loc ( Ω )(See Remark 2.6) and the assertions as in Lemma 2.9 and Lemma 2.12. Now using the Lemma3.1, we get z = u + u . That u = z − u is a solution to ( f P λ ). And from Lemma 4.1, we have u ∈ L ∞ ( Ω ). (cid:3) Lemma 4.5.
Let µ ≤ min { s, N } . Let u be any weak solution of problem ( P λ ) . Then u ∈ L ∞ ( Ω ) ∩ C + φ q ( Ω ) ∩ C γ ( R N ) where γ is defined (2.2) . Proof.
Let u be any weak solution of problem ( P λ ). Employing Lemma 4.4, u − u ∈ X is the solution to ( f P λ ) and which on taking account Lemma 6.3, we have u − u ∈ L ∞ ( Ω ).Therefore, u = ( u − u )+ u ∈ L ∞ ( Ω ). Let ˆ u be a unique solution (See [2, Theorem 1.2, Remark1.5]) to the following problem( − ∆ ) s ˆ u = ˆ u − q + λc, u > Ω, ˆ u = 0 in R N \ Ω where c = C ∗ | u | ∗ µ − ∞ with C ∗ = (cid:12)(cid:12)(cid:12)(cid:12) Z Ω dy | x − y | µ (cid:12)(cid:12)(cid:12)(cid:12) ∞ . Practising Lemma 3.1, one can easily showthat u ≤ u ≤ ˆ u a.e in Ω . Now using the fact that u ≤ u ≤ ˆ u a.e in Ω and regularity of u andˆ u we obtain u ∈ C + φ q ( Ω ). Observe that u is a classical solution in sense of [2, Definition 1] soby [2, Theorem 1.2], H¨older’s regularity follows. (cid:3) Proof of Theorem 2.14 : It follows from the proof of Lemma 4.5 and of Lemma 2.7. (cid:3) In this section, we have prove the existence of first solution and further establish that thefirst solution is actually a local minimizer of an appropriate functional. We start the sectionby defining the functional associated with ( f P λ ). Consider the functional J : X → ( −∞ , ∞ ]associated with J ( u ) = k u k + R Ω G ( x, u ) dx − λ ∗ µ RR Ω × Ω | u | ∗ µ | u | ∗ µ | x − y | µ dxdy if G ( · , u ) ∈ L ( Ω ) , ∞ otherwise.For any convex subset K ⊂ X we define the functional J K : X → ( −∞ , ∞ ] by J K ( u ) = ( J ( u ) if u ∈ K and G ( · , u ) ∈ L ( Ω ) , ∞ otherwise.Define Λ := sup { λ > P λ ) has a weak solution } . Lemma 5.1.
Let K be a convex subset of X and let w ∈ X . Let u ∈ K with G ( · , u ) ∈ L ( Ω ) .Then the following assertions are equivalent:(i) α ∈ ∂ − J K ( u ) .(ii) For every w ∈ K with G ( · , w ) ∈ L ( Ω ) , we have g ( · , u )( w − u ) ∈ L ( Ω ) and h α, w − u i ≤ h u, ( w − u ) i + Z Ω g ( x, u )( w − u ) dx − λ Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − ( w − u ) | x − y | µ dxdy. Proof. (i) implies (ii). Let w ∈ K and G ( · , w ) ∈ L ( Ω ). Define z = w − u . Thenclearly since g ( x, u ) is increasing in u , we have g ( x, u ) z ≤ G ( x, w ) − G ( x, u ). Moreover,( g ( · , u ) z ) ∨ ∈ L ( Ω ) and t ( G ( x, u + tz ) − G ( x, u )) /t, (0 , → R , is increasing and J K ( u + tz ) − J K ( u ) t = h u, w i + t k z k Z Ω ( G ( x, u + tz ) − G ( x, u )) t − ∗ µ t Z Z Ω × Ω ( u + u + tz ) ∗ µ ( u + u + tz ) ∗ µ | x − y | µ dxdy + 122 ∗ µ t Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ | x − y | µ dxdy. Passing to the limit as t → α ∈ ∂ − J K ( u ), we deduce the requiredresult. (ii) implies (i). Let z ∈ K and G ( · , w ) ∈ L ( Ω ). Employing the fact that G ( x, s ) is1convex is s and using (ii) we have that J K ( w ) − J K ( u ) = 12 k z k + Z Ω ( G ( x, w ) − G ( x, u ) − g ( x, u ) z ) dx + h α, z i− λ ∗ µ Z Z Ω × Ω (cid:0) ( w + u ) ∗ µ ( w + u ) ∗ µ − ( u + u ) ∗ µ ( u + u ) ∗ µ (cid:1) | x − y | µ dxdy + λ Z Z Ω × Ω ( u + u ) ∗ µ ( u + u ) ∗ µ − z | x − y | µ dxdy. It implies that α ∈ ∂ − J K ( u ). (cid:3) For any functions ϕ, ψ : Ω → [ −∞ , + ∞ ], we define the following subspaces K ϕ = { u ∈ X : ϕ ≤ u a.e } , K ψ = { u ∈ X : u ≤ ψ a.e } , K ψϕ = { u ∈ X : ϕ ≤ u ≤ ψ a.e } . Proposition 5.2.
Assume one the following condition holds:(i) φ is a subsolution to ( f P λ ) , G ( x, w ( x )) ∈ L loc ( Ω ) for all w ∈ K φ , u ∈ D ( J K φ ) and ∈ ∂ − J K φ ( u ) .(ii) φ is a supersolution to ( f P λ ) , G ( x, w ( x )) ∈ L loc ( Ω ) for all w ∈ K φ , u ∈ D ( J K φ ) and ∈ ∂ − J K φ ( u ) .(iii) φ , φ are subsolution and supersolution to ( f P λ ) , φ ≤ φ , G ( x, φ ) , G ( x, φ ) ∈ L loc ( Ω ) , u ∈ D ( J K φ φ ) and ∈ ∂ − J K φ φ ( u ) .Then u is weak solution to ( f P λ ) . Proof.
Follow the [21, Proposition 4.2], we have the required result. (cid:3)
Let ϑ ∈ C s ( R N ) ∩ X be the unique solution which satisfies ( − ∆ ) s ϑ = 1 / Ω in the senseof distributions. By the definition of g and G , we obtain the following properties Lemma 5.3. (i) Let u ∈ L loc ( Ω ) such that ess inf K u > for any compact set K ⊂ Ω .Then g ( x, u ( x )) , G ( x, u ( x )) ∈ L loc ( Ω ) .(ii) For all x ∈ Ω , the following holds(a) G ( x, st ) ≤ s G ( x, t ) for each s ≥ and t ≥ .(b) G ( x, s ) − G ( x, t ) − ( g ( x, s ) + g ( x, t ))( s − t ) / ≥ for each s, t with s ≥ t > − ϑ ( x ) .(c) G ( x, s ) − g ( x, s ) s/ ≥ for each s ≥ . Lemma 5.4.
The following hold:(i) is the strict subsolution to ( f P λ ) for all λ > .(ii) ϑ is a strict supersolution to ( f P λ ) for all sufficiently small λ > . (iii) Any positive weak solution w to ( g P λ ) is a strict supersolution to ( g P λ ) for < λ < λ . Proof. (i) Trivial.(ii) Choose λ small enough such that λ Z Ω ( ϑ + u ) ∗ µ | x − y | µ dy ! ( ϑ + u ) ∗ µ − < Ω . FromLemma 5.3, g ( x, ϑ ) , G ( x, ϑ ) ∈ L ( Ω ), for all ψ ∈ X \ { } , we deduce that h ϑ, ψ i + Z Ω g ( x, ϑ ) ψ dx − λ Z Ω Z Ω ( ϑ + u ) ∗ µ ( ϑ + u ) ∗ µ − ψ | x − y | µ dxdy ≥ Z − λ Z Ω ( ϑ + u ) ∗ µ | x − y | µ dy ! ( ϑ + u ) ∗ µ − ! ψ dx > . (iii) Let 0 < λ < λ and w be a positive weak solution to ( g P λ ). Then for all ψ ∈ X \ { } ,we have h w, ψ i + Z Ω g ( x, w ) ψ dx − λ Z Ω Z Ω ( w + u ) ∗ µ ( w + u ) ∗ µ − ψ | x − y | µ dxdy = ( λ − λ ) Z Ω Z Ω ( w + u ) ∗ µ ( w + u ) ∗ µ − ψ | x − y | µ dxdy > . The proof is now complete. (cid:3)
Theorem 5.5.
Let w , w : Ω → [ −∞ , + ∞ ] with w ≤ w such that w is a strict subsolutionto ( f P λ ) and u ∈ D ( J K w w ) be a minimizer for J K w w . Then u is a local minimizer for J K w . Proof.
For each v ∈ K w and 0 ≤ φ ∈ X , we define σ ( v ) = min { v, w } = v − ( v − w ) + and Ξ ( φ ) = h w , φ i + Z Ω g ( x, w ) φ dx − λ Z Ω Z Ω ( w + u ) ∗ µ ( w + u ) ∗ µ − φ | x − y | µ dxdy. Claim : h σ ( v ) , v − σ ( v ) i ≥ h w , v − σ ( v ) i and Z Ω Z Ω (cid:0) ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ − − ( w + u ) ∗ µ ( w + u ) ∗ µ − (cid:1) ( v − σ ( v )) | x − y | µ dxdy ≤ . Notice that v − σ ( v ) = ( v − w ) + . Let Ω = supp(( v − w ) + ). Then on Ω , σ ( v ) = w andusing the fact that σ ( v ) ≤ w on Ω , we have h σ ( v ) , v − σ ( v ) i = Z Ω Z Ω +2 Z R N \ Ω Z Ω + Z Ω \ Ω Z Ω \ Ω +2 Z R N \ Ω Z Ω \ Ω ( σ ( v )( x ) − σ ( v )( y ))(( v − σ ( v ))( x ) − ( v − σ ( v ))( y )) | x − y | N +2 s dxdy (cid:19) ≥ h w , v − σ ( v ) i . σ ( v ) ≤ w on Ω . It implies that the Claim holds. Takinginto account the fact that u is a minimizer of for J K w w , σ ( v ) ∈ D ( J K w w ), Lemma 2 of [29]and the fact that G ( x, · ) is convex, we infer that J K w ( v ) − J K w ( u ) ≥ J K w ( v ) − J K w ( σ ( v ))= k v − σ ( v ) k h σ ( v ) , v − σ ( v ) i + Z Ω ( G ( x, v ) − G ( x, σ ( v ))) dx − λ ∗ µ Z Ω Z Ω (cid:0) ( v + u ) ∗ µ ( v + u ) ∗ µ − ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ (cid:1) | x − y | µ dxdy ≥ k v − σ ( v ) k h σ ( v ) , v − σ ( v ) i + Z Ω g ( x, σ ( v ))( v − σ ( v )) dx − λ ∗ µ Z Ω Z Ω (cid:0) ( v + u ) ∗ µ ( v + u ) ∗ µ − ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ (cid:1) | x − y | µ dxdy ≥ k v − σ ( v ) k h w , v − σ ( v ) i + Z Ω g ( x, w )( v − σ ( v )) dx − λ ∗ µ Z Ω Z Ω (cid:0) ( v + u ) ∗ µ ( v + u ) ∗ µ − ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ (cid:1) | x − y | µ dxdy ≥ k v − σ ( v ) k Ξ ( v − σ ( v )) − λ ∗ µ I (5.1)where I = Z Ω Z Ω ( v + u ) ∗ µ ( v + u ) ∗ µ | x − y | µ dxdy − Z Ω Z Ω ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ | x − y | µ dxdy − ∗ µ Z Ω Z Ω ( σ ( v ) + u ) ∗ µ ( σ ( v ) + u ) ∗ µ − ( v − σ ( v )) | x − y | µ dxdy. Now we estimate I from above. First observe that I = 2 ∗ µ Z Ω Z vσ ( v ) Z Ω ( v + u ) ∗ µ + ( σ ( v ) + u ) ∗ µ | x − y | µ dy ! (cid:16) ( t + u ) ∗ µ − − ( σ ( v ) + u ) ∗ µ − (cid:17) dtdx + 2 ∗ µ Z Ω Z vσ ( v ) Z Ω ( v + u ) ∗ µ − ( σ ( v ) + u ) ∗ µ | x − y | µ dy ! ( σ ( v ) + u ) ∗ µ − dtdx. (5.2)4Using the mean value theorem, there exists θ ∈ [0 ,
1] such that( u + u ) ∗ µ − − ( v + u ) ∗ µ − ( u − v ) = (2 ∗ µ − u + u + θ ( v − u )) ∗ µ − ( u − v )= (2 ∗ µ − u + (1 − θ ) u + θv ) ∗ µ − ( u − v ) ≤ (2 ∗ µ − ∗ µ − ( u ∗ µ − + ((1 − θ ) u + θv ) ∗ µ − )( u − v ) ≤ (2 ∗ µ − ∗ µ − ( u ∗ µ − + max { u, v } ∗ µ − )( u − v ) . For each x ∈ Ω and v ∈ D ( J K w ) define the functions k v ( x ) = (2 ∗ µ − ∗ µ − ( u ∗ µ − + max {| w | , | v |} ∗ µ − ) χ { v>w } ,k v ( x ) = 2 ∗ µ ∗ µ − ( u ∗ µ − + max {| w | , | v |} ∗ µ − ) χ { v>w } . Using the Hardy-Littlewood-Sobolev inequality, we have Z Ω Z vσ ( v ) Z Ω ( v + u ) ∗ µ + ( σ ( v ) + u ) ∗ µ | x − y | µ dy ! (cid:16) ( t + u ) ∗ µ − − ( σ ( v ) + u ) ∗ µ − (cid:17) dtdx ≤ Z Ω Z Ω (( v + u ) ∗ µ + ( σ ( v ) + u ) ∗ µ ) k v ( x )( v − σ ( v )) | x − y | µ dydx ≤ c (cid:16) | v + u | ∗ µ ∗ s + | σ ( v ) + u | ∗ µ ∗ s (cid:17) | k v ( x )( v − σ ( v )) | ∗ s ∗ µ (5.3)for some appropriate positive constant c . Similarly with the help of Hardy-Littlewood-Sobolev inequality, H¨older’s inequality and the definition of S we have Z Ω Z vσ ( v ) Z Ω ( v + u ) ∗ µ − ( σ ( v ) + u ) ∗ µ | x − y | µ dy ! ( σ ( v ) + u ) ∗ µ − dtdx ≤ c S − / | k v ( x )( v − σ ( v )) | ∗ s ∗ µ | σ ( v ) + u | ∗ µ − ∗ s k v − σ ( v ) k (5.4)for some appropriate positive constant c . Using (5.2) jointly with (5.3) and (5.4), we have I ≤ c (cid:16) | v + u | ∗ µ ∗ s + | σ ( v ) + u | ∗ µ ∗ s (cid:17) | k v ( x )( v − σ ( v )) | ∗ s ∗ µ + c S − / | k v ( x )( v − σ ( v )) | ∗ s ∗ µ | σ ( v ) + u | ∗ µ − ∗ s k v − σ ( v ) k . (5.5)Let us suppose that the result is not true. This means that there exists a sequence { v n } ⊂ X such that for any v n ∈ K w and k v n − u k < n , J K w ( v n ) < J K w ( u ) for all n. l := u + P ∞ n =1 | v n − u | . By definition, | v n | ≤ l a.e for all n . Now for each v ∈ D ( J K w ),set k v ( x ) = (2 ∗ µ − ∗ µ − ( u ∗ µ − + max {| w | , | l |} ∗ µ − ) χ { v>w } k v ( x ) = 2 ∗ µ ∗ µ − ( u ∗ µ − + max {| w | , | l |} ∗ µ − ) χ { v>w } . Employing (5.1) and (5.5), we deduce that0 > J K w ( v n ) − J K w ( u ) ≥ J K w ( v n ) − J K w ( σ ( v n )) ≥ k v n − σ ( v n ) k − λ (cid:18) c (cid:16) | v n + u | ∗ µ ∗ s + | σ ( v n ) + u | ∗ µ ∗ s (cid:17) | k v n ( x )( v n − σ ( v n )) | ∗ s ∗ µ + c S − / | k v n ( x )( v n − σ ( v n )) | ∗ s ∗ µ | σ ( v n ) + u | ∗ µ − ∗ s k v n − σ ( v n ) k (cid:19) + Ξ ( v n − σ ( v n )) ≥ k v n − σ ( v n ) k Ξ ( v n − σ ( v n )) − (cid:18) C | k v n ( x )( v n − σ ( v n )) | ∗ s ∗ µ + C | k v n ( x )( v n − σ ( v n )) | ∗ s ∗ µ k v n − σ ( v n ) k (cid:19) (5.6)where C = sup n λc (cid:16) | v n + u | ∗ µ ∗ s + | σ ( v n ) + u | ∗ µ ∗ s (cid:17) and C = sup n λc S − / | σ ( v n ) + u | ∗ µ − ∗ s .Consider | k v n ( x )( v n − σ ( v n )) | ∗ s ∗ µ ≤ | k v n ( x ) | ∗ s ∗ µ − | ( v n − σ ( v n )) | ∗ s ∗ µ = Z { k vn ≤ R } | k v n ( x ) | ∗ s ∗ µ − ! ∗ µ − ∗ s + Z { k vn >R } | k v n ( x ) | ∗ s ∗ µ − ! ∗ µ − ∗ s | ( v n − σ ( v n )) | ∗ s ∗ µ . Choose R , R > n , C S − Z { k vn >R } | k v n ( x ) | ∗ s ∗ µ − ! ∗ µ − ∗ s <
12 and C S − / Z { k vn >R } | k v n ( x ) | ∗ s ∗ µ − ! ∗ µ − ∗ s < . > k v n − σ ( v n ) k Ξ ( v n − σ ( v n )) − C R (cid:18)Z Ω ( v n − σ ( v n )) ∗ s ∗ µ dx (cid:19) ∗ µ ∗ s + C R (cid:18)Z Ω ( v n − σ ( v n )) ∗ s ∗ µ dx (cid:19) ∗ µ ∗ s k v n − σ ( v n ) k ≥ k ( v n − w ) + k Ξ (( v n − w ) + ) − C R | ( v n − w ) + | ∗ s ∗ µ + C R | ( v n − w ) + | ∗ s ∗ µ k v n − σ ( v n ) k ! . Let C ∗ = max { C R , C R } . Thus0 > k ( v n − w ) + k Ξ (( v n − w ) + ) − C ∗ | ( v n − w ) + | ∗ s ∗ µ + | ( v n − w ) + | ∗ s ∗ µ k ( v n − w ) + k ! . (5.7)Let ν = inf { Ξ ( φ ) : φ ∈ A} where A = { φ ∈ X : φ ≥ , | φ | ∗ s ∗ µ = 1 , k φ k ≤ C ∗ } .Clearly, A is a weakly sequentially closed subset of X . Using Fatou’s lemma and the factthat Riesz potential is a bounded linear functional, one can easily prove that Ξ is a weaklylower semicontinuous on A . Hence ν >
0. Indeed, let z n is a minimizing sequence of ν suchthat z n ⇀ z weakly in X as n → ∞ then Ξ ( z ) ≤ lim inf Ξ ( z n ). Now by the application ofthe fact that w is a strict supersolution to ( f P λ ) we get that Ξ ( z ) >
0. Now notice that usingthe definition of ν , (5.7) can be rewritten as the following0 >ν + 14 (cid:18) k ( v n − w ) + k − C ∗ | ( v n − w ) + | ∗ s ∗ µ (cid:19) − (( C ∗ ) + 2 C ∗ ) | ( v n − w ) + | ∗ s ∗ µ ! > ν −
14 (( C ∗ ) + 2 C ∗ ) | ( v n − w ) + | ∗ s ∗ µ (5.8)As v n is a sequence such that v n → u in X . It implies that as n → ∞ , | ( v n − w ) + | ∗ s ∗ µ → ν >
0. Hence the proof is complete. (cid:3)
Lemma 5.6.
Λ > . Proof.
We will use the lower and upper solution method to prove the required result. FromLemma 5.4, 0 and ϑ are the sub and supersolution respectively to ( f P λ ). We define the closed7convex set of X as W = { u ∈ X : 0 ≤ u ≤ ϑ } . Employing the definition of W , one can easily prove that J W ≥ k u k − c − c for appropriate positive constants c and c . It implies J W is coercive on W . J W is weaklylower semi continuous on W . Indeed, let { u n } ⊂ W such that u n ⇀ u weakly in X as n → ∞ . For each n , Z Ω G ( x, u n ) dx ≤ Z Ω G ( x, u ) dx < + ∞ , Z Z Ω × Ω ( u n + u ) ∗ µ ( u n + u ) ∗ µ | x − y | µ dxdy ≤ Z Z Ω × Ω ( ϑ + u ) ∗ µ ( ϑ + u ) ∗ µ | x − y | µ dxdy < + ∞ . Now we may use the dominated convergence theorem and the weak lower semicontinuity ofthe norm to prove that J W is weakly lower semi continuous on W . Thus, there exists u ∈ X such that inf v ∈ W J W ( v ) = J W ( u ) . Since 0 ∈ ∂ − J W ( u ), u is a weak solution to ( f P λ ). It implies Λ > (cid:3) Theorem 5.7.
Let λ ∈ (0 , Λ ) . Then there exists a positive weak solution u λ to ( f P λ ) belongingto X such that J ( u λ ) < and u λ is a local minimizer for J K . Proof.
Let λ ∈ (0 , Λ ) and λ ′ ∈ ( λ, Λ ). Then by Lemma 5.4, 0 and u λ ′ are strict sub andsupersolution to ( f P λ ). The existence of u λ ′ is clear by the definition of Λ . Consider the convexset W = { u ∈ X : 0 ≤ u ≤ u λ ′ } . Using the same analysis as in Lemma 5.6, there exists a u λ ∈ X such thatinf v ∈ W J W ( v ) = J W ( u λ ). Notice that 0 ∈ W and J W (0) <
0, it impliesthat J W ( u λ ) <
0. Let φ = 0 and φ = u λ ′ in Theorem 5.5 we have u λ is a local minimizerof J K . (cid:3) Lemma 5.8.
Λ < ∞ . Proof.
Assume by contradiction that Λ = + ∞ . Then there exists a sequence λ n → ∞ as n → ∞ . Let u λ n be the corresponding solution to ( f P λ ). Then by Theorem 5.7, J ( u λ n ) < u λ n is a local minimizer for J K . That is,12 k u λ n k + Z Ω G ( x, u λ n ) dx − λ n ∗ µ Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ | x − y | µ dxdy < k u λ n k + Z Ω g ( x, u λ n ) u λ n dx − λ n Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − u λ n | x − y | µ dxdy = 0 . (5.9)With the application of Lemma 5.3(ii) and statements, we have12 Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − u λ n | x − y | µ dxdy < ∗ µ Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ | x − y | µ dxdy. (5.10)Using the fact that u ∈ L ∞ ( Ω ), for each x ∈ Ω , we havelim t →∞ (cid:18)R Ω | t + u | ∗ µ | x − y | µ dy (cid:19) | t + u | ∗ µ (cid:18)R Ω | t + u | ∗ µ | x − y | µ dy (cid:19) | t + u | ∗ µ − t = 1 . Therefore, it follows that for any small enough ε >
0, there exists M ε > n ∗ µ Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ | x − y | µ dxdy<
12 + ε Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − u λ n | x − y | µ dxdy + M ε . (5.11)From (5.10) and (5.11), we obtain Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − u λ n | x − y | µ dxdy < ∞ for all n. From (5.9), we have k u λ n k < λ n Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − u λ n | x − y | µ dxdy. Hence { λ − / n u λ n } is uniformly bounded in X . Then there exists z ∈ X such that z n := λ − / n u λ n ⇀ z weakly in X . Let 0 ≤ ψ ∈ C ∞ c ( Ω ) be a non trivial function. Let k > u > k on supp( ψ ). Once again using (5.9), we deduce that p λ n Z Z Ω × Ω k ∗ µ − ψ | x − y | µ dxdy ≤ p λ n Z Z Ω × Ω ( u λ n + u ) ∗ µ ( u λ n + u ) ∗ µ − ψ | x − y | µ dxdy = h z n , ψ i + 1 √ λ n Z Ω g ( x, u λ n ) ψ dx ≤ h z n , ψ i + 1 √ λ n Z Ω k − q ψ dx. Now passing the limit n → ∞ , we have h z , ψ i = ∞ , which is not true. Hence Λ < ∞ . (cid:3) In this section we will prove the existence of second solution to ( f P λ ). Let u λ denotes the firstsolution to ( f P λ ) obtained in Theorem 5.7. Proposition 6.1.
The functional J K uλ satisfies the ( CP S ) c for each c satisfying c < J K uλ ( u λ ) + 12 (cid:18) N − µ + 2 s N − µ (cid:19) S N − µN − µ +2 s H,L λ N − sN − µ +2 s . Proof.
Let c < J K uλ ( u λ ) + (cid:16) N − µ +2 s N − µ (cid:17) S N − µN − µ +2 sH,L λ N − sN − µ +2 s . Let z n be a sequence such that J K uλ ( z n ) → c and (1 + k z n k ) ||| ∂ − J K uλ ( z n ) ||| → n → ∞ . It implies there exists ξ n ∈ ∂ − J K uλ ( z n ) such that k ξ n k = ||| ∂ − J K uλ ( z n ) ||| for every n . FromLemma 5.1, for each v ∈ D ( J K uλ ) and for each n , g ( · , z n )( v − z n ) ∈ L ( Ω ) and h ξ n , v − z n i ≤ h z n , v − z n i + Z Ω g ( x, z n )( v − z n ) dx − λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( v − z n ) | x − y | µ dxdy. (6.1)Using the fact that G ( · , z n ) ∈ L ( Ω ) and Lemma 5.3, we obtain that G ( · , z n ) ∈ L ( Ω ). So2 z n ∈ D ( J K uλ ), now employing (6.1), we get h ξ n , z n i ≤ k z n k + Z Ω g ( x, z n ) z n dx − λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ − z n | x − y | µ dxdy. ε > c + 1 ≥ k z n k + Z Ω G ( x, z n ) dx − λ ∗ µ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ | x − y | µ dxdy ≥ k z n k + Z Ω G ( x, z n ) dx −
12 + ε (cid:18) h ξ n , z n i − k z n k − Z Ω g ( x, z n ) z n dx (cid:19) − λM ε ≥ k z n k −
12 + ε (cid:0) h ξ n , z n i − k z n k (cid:1) − λM ε . It shows that { z n } is a bounded sequence in X . Hence, up to a subsequence, there ex-ist z ∈ X such that z n ⇀ z weakly in X as n → ∞ . Let k z n − z k → a and RR Ω × Ω ( z n − z ) ∗ µ ( z n − z ) ∗ µ | x − y | µ dxdy → b ∗ µ as n → ∞ . By the mean value theorem, Brezis-LiebLemma (see [7, 17]) and (6.1), we deduce that Z Ω G ( x, z ) dx ≥ Z Ω G ( x, z n ) dx + Z Ω g ( x, z n )( z − z n ) dx ≥ Z Ω G ( x, z n ) dx − λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( z n − z ) | x − y | µ dxdy − h ξ n , z n − z i + h z n , z n − z i = Z Ω G ( x, z n ) dx − h ξ n , z n − z i + h z n , z n − z i− λ Z Z Ω × Ω (cid:0) ( z n − z ) ∗ µ ( z n − z ) ∗ µ + ( z + u ) ∗ µ ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy + λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( z + u ) | x − y | µ dxdy. Now using the fact that z n converges to z weakly in X , hence as n → ∞ , we get Z Ω G ( x, z ) dx ≥ Z Ω G ( x, z ) dx + a − λb ∗ µ . Thus λb ∗ µ ≥ a . (6.2)On account of the fact that u λ is a weak positive solution to ( f P λ ), for each n ,0 = h u λ , z n − u λ i + Z Ω g ( x, u λ )( z n − u λ ) dx + λ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n − u λ ) | x − y | µ dxdy. (6.3)In consideration of G ( · , z n ) , G ( · , z n ) ∈ L ( Ω ) and u λ ≤ z n − u λ ≤ z n , we infer that12 z n − u λ ∈ D ( J K uλ ). Testing (6.1) with 2 z n − u λ , we obtain that h ξ n , z n − u λ i ≤ h z n , z n − u λ i + Z Ω g ( x, z n )( z n − u λ ) dx − λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( z n − u λ ) | x − y | µ dxdy. (6.4)From Lemma 5.3, (6.3) and (6.4), we have J K uλ ( z n ) − J K uλ ( u λ )= 12 k z n k + Z Ω G ( x, z n ) dx − λ ∗ µ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ | x − y | µ dxdy − k u λ k − Z Ω G ( x, u λ ) dx + λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ | x − y | µ dxdy ≥ Z Ω (cid:18) G ( x, z n ) − G ( x, u λ ) −
12 ( g ( x, u λ ) + g ( x, z n )) ( z n − u λ ) (cid:19) dx + λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n + u ) ∗ µ ( z n + u ) ∗ µ | x − y | µ dxdy + 12 h ξ n , z n − u λ i + λ Z Z Ω × Ω (cid:0) ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − − ( z n + u ) ∗ µ ( z n + u ) ∗ µ − (cid:1) ( z n − u λ ) | x − y | µ dxdy ≥ λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n + u ) ∗ µ ( z n + u ) ∗ µ | x − y | µ dxdy + λ Z Z Ω × Ω (cid:0) ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n − u λ ) − ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( u λ + u ) (cid:1) | x − y | µ dxdy + λ Z Z Ω × Ω ( z n + u ) ∗ µ ( z n + u ) ∗ µ | x − y | µ dxdy + 12 h ξ n , z n − u λ i =: I + 12 h ξ n , z n − u λ i . (6.5)Using Brezis-Lieb Lemma (See [17]), we have I = λ (cid:18) − ∗ µ (cid:19) Z Z Ω × Ω (cid:0) ( z n − z ) ∗ µ ( z n − z ) ∗ µ (cid:1) + (cid:0) ( z + u ) ∗ µ ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy + λ Z Z Ω × Ω (cid:0) ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n − u λ ) (cid:1) − (cid:0) ( z n + u ) ∗ µ ( z n + u ) ∗ µ − ( u λ + u ) (cid:1) | x − y | µ dxdy + λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ | x − y | µ dxdy + o (1) . (6.6)2Observe that by weak convergence of the sequence { z n } , we have Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z n − z ) | x − y | µ dxdy → Z Z Ω × Ω (cid:0) ( z n + u ) ∗ µ ( z n + u ) ∗ µ − − ( z + u ) ∗ µ ( z + u ) ∗ µ − (cid:1) ( u λ + u ) | x − y | µ dxdy → . (6.7)Taking into account (6.5), (6.6), (6.7) and passing the limit as n → ∞ , we obtain that c − J K uλ ( u λ ) ≥ λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ | x − y | µ dxdy + λ (cid:18) − ∗ µ (cid:19) b ∗ µ + λ Z Z Ω × Ω ( u λ + u ) ∗ µ ( u λ + u ) ∗ µ − ( z − u λ ) | x − y | µ dxdy + λ Z Z Ω × Ω ( z + u ) ∗ µ ( z + u ) ∗ µ − ( z − u λ ) | x − y | µ dxdy − λ ∗ µ Z Z Ω × Ω ( z + u ) ∗ µ ( z + u ) ∗ µ | x − y | µ dxdy := I (say) + λ (cid:18) − ∗ µ (cid:19) b ∗ µ . (6.8)Clearly, we infer I = λ Z Z Ω × Ω ( u λ + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − + ( z + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy − λ Z Z Ω × Ω ( u λ + u ) ∗ µ ( z + u ) ∗ µ − ( z − u λ ) | x − y | µ dxdy + λ Z Z Ω × Ω ( z + u ) ∗ µ (cid:0) ( z + u ) ∗ µ − + ( u λ + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy − λ Z Z Ω × Ω ( z + u ) ∗ µ ( u λ + u ) ∗ µ − ( z − u λ ) | x − y | µ dxdy + λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy + λ ∗ µ Z Z Ω × Ω ( z + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy. (6.9)Since ( u λ + u ) ∗ µ − ( z + u ) ∗ µ = − ∗ µ Z z u λ ( t + u ) ∗ µ − dt ≥ − ∗ µ ( u λ + u ) ∗ µ − + ( z + u ) ∗ µ − ! ( z − u λ ) . λ ∗ µ Z Z Ω × Ω ( u λ + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy ≥ − λ Z Z Ω × Ω ( u λ + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − + ( z + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy. (6.10)Similarly, λ ∗ µ Z Z Ω × Ω ( z + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − ( z + u ) ∗ µ (cid:1) | x − y | µ dxdy ≥ − λ Z Z Ω × Ω ( z + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − + ( z + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy. (6.11)From (6.9), (6.10) and (6.11), we deduce that I = λ Z Z Ω × Ω ( u λ + u ) ∗ µ (cid:0) ( u λ + u ) ∗ µ − − ( z + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy + λ Z Z Ω × Ω ( z + u ) ∗ µ (cid:0) ( z + u ) ∗ µ − − ( u λ + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy = λ Z Z Ω × Ω (cid:0) ( z + u ) ∗ µ − ( u λ + u ) ∗ µ (cid:1) (cid:0) ( z + u ) ∗ µ − − ( u λ + u ) ∗ µ − (cid:1) ( z − u λ ) | x − y | µ dxdy ≥ . (6.12)Hence from (6.8) and (6.12), we obtain c − J K uλ ( u λ ) ≥ λ (cid:18) − ∗ µ (cid:19) b ∗ µ . (6.13)Using definition of S H,L and (6.2), we have λb ∗ µ ≥ a and a ≥ S H,L b , that is b ≥ (cid:18) S H,L λ (cid:19) N − s N − µ +2 s ) . (6.14)Using (6.13) and (6.14), we get c − J K uλ ( u λ ) ≥ λ (cid:18) − ∗ µ (cid:19) (cid:18) S H,L λ (cid:19) N − µN − µ +2 s = 12 (cid:18) N − µ + 2 s N − µ (cid:19) S N − µN − µ +2 s H,L λ N − sN − µ +2 s . It contradicts the fact that c < J K uλ ( u λ ) + (cid:16) N − µ +2 s N − µ (cid:17) S N − µN − µ +2 sH,L λ N − sN − µ +2 s . Hence a = 0. (cid:3) { U ε } ε> of S defined as U ε = ε − ( N − s )2 S ( N − µ )(2 s − N )4( N − µ +2 s ) ( C ( N, µ )) s − N N − µ +2 s ) u ∗ ( x/ε )where u ∗ ( x ) = u ( x/S / s ) , u ( x ) = ˜ u ( x ) | ˜ u | ∗ s and ˜ u ( x ) = a ( b + | x | ) − ( N − s )2 with α ∈ R \ { } and β > ε > , U ε satisfies( − ∆ ) s u = ( | x | − µ ∗ | u | ∗ µ ) | u | ∗ µ − u in R N . Let ̺ > B ̺ ⊂ Ω . Now define η ∈ C ∞ c ( R N ) such that 0 ≤ η ≤ R N , η ≡ B ̺ (0) and η ≡ R N \ B ̺ (0). For each ε > x ∈ R N , we define Ψ ε = η ( x ) U ε ( x ). Proposition 6.2.
Let
N > s, < µ < N then the following holds:(i) k Ψ ε k ≤ S N − µN − µ +2 s H,L + O ( ε N − s ) .(ii) k Ψ ε k . ∗ µ NL ≤ S N − µN − µ +2 s H,L + O ( ε N ) .(iii) k Ψ ε k . ∗ µ NL ≥ S N − µN − µ +2 s H,L − O ( ε N ) . Proof.
Using the definition of Ψ ε and [38, Proposition 1] part ( i ) follows. For ( ii ) and ( iii )see [22, Proposition 2.8]. (cid:3) Lemma 6.3. [25] The following holds:(i) If µ < min { s, N } then for all Θ < , k u λ + tΨ ε k . ∗ µ NL ≥ k u λ k . ∗ µ NL + k Ψ ε k . ∗ µ NL + b Ct . ∗ µ − Z Ω Z Ω ( Ψ ε ( x )) ∗ µ ( Ψ ε ( y )) ∗ µ − u λ ( y ) | x − y | µ dxdy + 2 . ∗ µ t Z Ω Z Ω ( u λ ( x )) ∗ µ ( u λ ( y )) ∗ µ − Ψ ε ( y ) | x − y | µ dxdy − O ( ε ( N − µ ) Θ ) . (ii) There exists a R > such that R Ω R Ω ( Ψ ε ( x )) ∗ µ ( Ψ ε ( y )) ∗ µ − u λ ( y ) | x − y | µ dxdy ≥ b CR ε N − s . Lemma 6.4. sup {J K uλ ( u λ + tΨ ε ) : t ≥ } < J K uλ ( u λ ) + (cid:16) N − µ +2 s N − µ (cid:17) S N − µN − µ +2 sH,L λ N − sN − µ +2 s for anysufficiently small ε > . Proof.
Employing the fact that u λ is a weak solution to ( P λ ) and using Lemma 6.3, for all5 Θ <
1, we have J K uλ ( u λ + tΨ ε ) − J K uλ ( u λ ) ≤ k tΨ ε k − λ ∗ µ k tΨ ε k . ∗ µ NL + O ( ε ( N − µ ) Θ )+ Z Ω ( G ( u λ + tΨ ε ) − G ( x, u λ ) − g ( x, u λ ) tΨ ε ) dx − λ b Ct . ∗ µ − ∗ µ Z Ω Z Ω ( Ψ ε ( x )) ∗ µ ( Ψ ε ( y )) ∗ µ − u λ ( y ) | x − y | µ dxdy. From Proposition 6.2 and Lemma 6.3, we deduce that J K uλ ( u λ + tΨ ε ) − J K uλ ( u λ ) ≤ t S N − µN − µ +2 s H,L + O ( ε N − s )) − λt ∗ µ ∗ µ ( S N − µN − µ +2 s H,L − O ( ε N ))+ Z Ω ( G ( u λ + tΨ ε ) − G ( x, u λ ) − g ( x, u λ ) tΨ ε ) dx − b Ct . ∗ µ − ∗ µ b CR ε N − s + O ( ε ( N − µ ) Θ ) . (6.15)Observe that for any fix 1 < ρ < min { , n − s } , there exists R > Z Ω | Ψ ε | ρ dx ≤ R ε ( n − s ) ρ/ . Moreover, there exists R > x ∈ Ω, r > m and s ≥ G ( x, r + s ) − G ( x, s ) − g ( x, r ) s = Z r + sr ( τ − q − r − q ) dτ ≤ R s ρ . Using last inequality and (6.15) with Θ = ∗ µ , we obtain J K uλ ( u λ + tΨ ε ) − J K uλ ( u λ ) ≤ t S N − µN − µ +2 s H,L + O ( ε N − s )) − t ∗ µ ∗ µ ( S N − µN − µ +2 s H,L − O ( ε N )) − b Ct . ∗ µ − ∗ µ b CR ε N − s + R R t ρ ε ( n − s ) ρ/ + o ( ε N − s ):= K ( t ) . Clearly, one can check that K ( t ) → −∞ , K ( t ) > t → + and there exists t ε > K ′ ( t ε ) = 0. Furthermore, there exist positive constants T and T such that T ≤ t ε ≤ T (for6details see [25]). Hence, K ( t ) ≤ t ε S N − µN − µ +2 s H,L + O ( ε N − s )) − t ∗ µ ε ∗ µ ( S N − µN − µ +2 s H,L − O ( ε N )) − b CT . ∗ µ − ∗ µ b CR ε N − s + R R T ρ ε ( n − s ) ρ/ + o ( ε N − s ) ≤ sup t ≥ K ( t ) − b CT . ∗ µ − ∗ µ b CR ε N − s + R R T ρ ε ( n − s ) ρ/ + o ( ε N − s )where K ( t ) = t ( S N − µN − µ +2 s H,L + O ( ε N − s )) − t ∗ µ ∗ µ ( S N − µN − µ +2 s H,L − O ( ε N )). By trivial computations,we get J K uλ ( u λ + tΨ ε ) − J K uλ ( u λ ) ≤ (cid:18) N − µ + 2 s N − µ (cid:19) S N − µN − µ +2 s H,L λ N − sN − µ +2 s + O ( ε N − s ) − Cε N − s + o ( ε N − s )for an appropriate constant C >
0. Thus, for ε sufficiently small, J K uλ ( u λ + tΨ ε ) − J K uλ ( u λ ) < (cid:18) N − µ + 2 s N − µ (cid:19) S N − µN − µ +2 s H,L λ N − sN − µ +2 s . Hence the proof follows. (cid:3)
Proposition 6.5.
For each λ ∈ (0 , Λ ) there exist a second positive solution to ( f P λ ) . Proof.
From Theorem 5.7, u λ is a local minimizer of J K uλ . It implies there exist ς > J K uλ ( z ) ≥ J K uλ ( u λ ) for every z ∈ K u λ with k z − u λ k ≤ ς . Let Ψ = Ψ ε for ε obtained in Lemma 6.4. Since J K uλ ( u λ + tΨ ) → −∞ as t → ∞ , so choose t ≥ ς/ k Ψ k suchthat J K uλ ( u λ + tΨ ) ≤ J K uλ ( u λ ). Define Σ = { φ ∈ C ([0 , , D ( J K uλ )) : φ (0) = u λ , φ (1) = u λ + tΨ } ,A = { z ∈ D ( J K uλ ) : k z − u λ k = α } and c = inf φ ∈ Σ sup r ∈ [0 , J K uλ ( φ ( r )) . With the help of Proposition 6.1 and Lemma 6.4, J K uλ satisfies (CPS) c condition. If c = J K uλ ( u λ ) = inf J K uλ ( A ) then u λ A, u λ + tΨ A, inf J K uλ ( A ) ≥ J K uλ ( u λ ) ≥ J K uλ ( u λ + tΨ ), and for every φ ∈ Σ , there exist r ∈ [0 ,
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